The number of ways in which $5$ beads of different colours form a necklace is

In how many ways can a team of $10$ players be selected from $22$ players if $6$ particular players are always to be included and $4$ particular players are always excluded?

The number of ways in which an arrangement of $4$ letters of the word $PROPORTION$ can be made is

Find $n$ if ${ }^{n} P_{4}=18 \cdot{ }^{n-1} P_{2}$.

For a set of $5$ true-or-false questions,no student has written all the correct answers,and no $2$ students have given the same sequence of answers. What is the maximum number of students in the class,for this to be possible?

$4$ alphabets $E, K, S$ and $V$,one in each,were purchased from a plastic warehouse. How many ordered pairs of alphabets,to be used as initials,can be formed from them?